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Mirrors > Home > MPE Home > Th. List > imass1 | Structured version Visualization version GIF version |
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.) |
Ref | Expression |
---|---|
imass1 | ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssres 5968 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶)) | |
2 | rnss 5898 | . . 3 ⊢ ((𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶) → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ran (𝐴 ↾ 𝐶) ⊆ ran (𝐵 ↾ 𝐶)) |
4 | df-ima 5650 | . 2 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
5 | df-ima 5650 | . 2 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶) | |
6 | 3, 4, 5 | 3sstr4g 3993 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 “ 𝐶) ⊆ (𝐵 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3914 ran crn 5638 ↾ cres 5639 “ cima 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-cnv 5645 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 |
This theorem is referenced by: predrelss 6295 vdwnnlem1 16875 dprdres 19815 imasnopn 23064 imasncld 23065 imasncls 23066 utoptop 23609 restutop 23612 ustuqtop3 23618 utopreg 23627 metustbl 23945 imadifxp 31572 esum2d 32756 eulerpartlemmf 33039 bj-imdirco 35711 brtrclfv2 42091 frege97d 42116 frege109d 42121 frege131d 42128 hess 42144 resimass 43557 setrecsss 47236 |
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